Intermediate Statistics Research Articles

Peak-over-threshold estimators for spectral tail processes: random vs deterministic thresholds

The extreme value dependence of regularly varying stationary time series can be described by the spectral tail process. Drees et al. (Extremes 18(3), 369–402, 2015) proposed estimators of the marginal distributions of this process based on exceedances over high deterministic thresholds and analyzed their asymptotic behavior. In practice, however, versions of the estimators are applied which use exceedances over random thresholds like intermediate order statistics. We prove that these modified estimators have the same limit distributions. This finding is corroborated in a simulation study, but the version using order statistics performs a bit better for finite samples.

Eigenvalue statistics for generalized symmetric and Hermitian matrices

Random matrix theory predicts the level statistics of a Hamiltonian to exhibit either clustering or repulsion if the underlying dynamics is integrable or chaotic, respectively. In various physical systems it is also possible to observe intermediate spectral properties showing the transition between different symmetry classes. In this work, we study generalized random matrix ensembles by dropping the constraint of canonical invariance and considering different variances in the diagonal and off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit intermediate level statistics. The nearest neighbour spacing (NNS) distributions can be computed for generalized symmetric matrices exhibiting crossover from clustering to GOE-like repulsion. The analysis is extended to matrices where the distributions of NNS as well as ratio of nearest neighbour spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices () where crossovers also take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes to quantify the system size dependence of the crossover boundary.

Experimental investigation of the fluctuations in nonchaotic scattering in microwave billiards**Project supported by the National Natural Science Foundation of China (Grant Nos. 11775100, 11775101, and 11961131009).

We report on the experimental investigation of the properties of the eigenvalues and wavefunctions and the fluctuation properties of the scattering matrix of closed and open billiards, respectively, of which the classical dynamics undergoes a transition from integrable via almost integrable to fully chaotic. To realize such a system, we chose a billiard with a 60° sector shape of which the classical dynamics is integrable, and introduced circular scatterers of varying number, size, and position. The spectral properties of generic quantum systems of which the classical counterpart is either integrable or chaotic are universal and well understood. If, however, the classical dynamics is pseudo-integrable or almost-integrable, they exhibit a non-universal intermediate statistics, for which analytical results are known only in a few cases, e.g., if it corresponds to semi-Poisson statistics. Since the latter is, above all, clearly distinguishable from those of integrable and chaotic systems, our aim was to design a billiard with these features which indeed is achievable by adding just one scatterer of appropriate size and position to the sector billiard. We demonstrated that, while the spectral properties of almost-integrable billiards are sensitive to the classical dynamics, this is not the case for the distribution of the wavefunction components, which was analyzed in terms of the strength distribution, and the fluctuation properties of the scattering matrix which coincide with those of typical, fully chaotic systems.

Spectral properties of quantum circulant graphs

We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain important features of the prototypical quantum star graph model. In particular, we show the spectrum is encoded in a secular equation with similar features. The secular equation of a quantum circulant graph takes two forms depending on whether the edge lengths respect the cyclic symmetry of the graph. When all the edge lengths are incommensurate, the spectral statistics correspond to those of random matrices from the Gaussian orthogonal ensemble according to the conjecture of Bohigas, Giannoni and Schmit. When the edge lengths respect the cyclic symmetry the spectrum decomposes into subspectra whose corresponding eigenfunctions transform according to irreducible representations of the cyclic group. We show that the subspectra exhibit intermediate spectral statistics and analyze the small and large parameter asymptotics of the two-point correlation function, applying techniques developed from star graphs. The particular form of the intermediate statistics differs from that seen for star graphs or Dirac rose graphs. As a further application, we show how the secular equations can be used to obtain spectral zeta functions using a contour integral technique. Results for the spectral determinant and vacuum energy of circulant graphs are obtained from the zeta functions.

Level statistics across the many-body localization transition

Level statistics of systems that undergo many--body localization transition are studied. An analysis of the gap ratio statistics from the perspective of inter- and intra-sample randomness allows us to pin point differences between transitions in random and quasi-random disorder, showing the effects due to Griffiths rare events for the former case. It is argued that the transition in the case of random disorder exhibits universal features that are identified by constructing an appropriate model of intermediate spectral statistics which is a generalization of the family of short-range plasma models. The considered weighted short-range plasma model yields a very good agreement both for level spacing distribution including its exponential tail and the number variance up to tens of level spacings outperforming previously proposed models. In particular, our model grasps the critical level statistics which arise at disorder strength for which the inter-sample fluctuations are the strongest. Going beyond the paradigmatic examples of many-body localization in spin systems, we show that the considered model also grasps the level statistics of disordered Bose- and Fermi-Hubbard models. The remaining deviations for long-range spectral correlations are discussed and attributed mainly to the intricacies of level unfolding.

A Comparison of Finite-Rate Kinetics and Flamelet-Generated Manifold Using a Multiscale Modeling Framework for Turbulent Premixed Combustion

ABSTRACTTurbulent premixed flames observed in combustion devices exhibit the presence of multiscale and nonlinear turbulence–chemistry interactions, which make predictions using large-eddy simulation (LES) extremely challenging. In this study, two well-established chemistry modeling paradigms, namely, finite-rate chemistry (FRC) and flamelet generated manifold (FGM) are compared using the same computational framework to assess their predictive capabilities. The classical FGM-based framework for LES is extended further by using it in the linear eddy mixing (LEM) model for the closure of the filtered source term for the scalar field. The assessment is performed using two computational setups, a canonical configuration of a freely propagating methane-air turbulent premixed flame, and a practically relevant swirl premixed combustor, both in the thin reaction zone regime, where the focus is to examine the effects of chemistry modeling paradigm and SGS turbulence–chemistry interactions. While for the first setup, direct numerical simulation (DNS) is considered as a reference for comparison of different models, for the second configuration, the results are compared with the available experimental data. Overall, while mean flow features are predicted reasonably well by the FGM-based approaches, quantitative differences are observed for intermediate species and conditional statistics when compared with the FRC-based approaches, and possible reasons for these differences are discussed.

An Updated Recommendation for Multiple Comparisons

Instructors of introductory and intermediate statistics courses often teach the use of analysis of variance (ANOVA) for the purpose of comparing more than two group means and pairwise comparison procedures (PCPs) to determine which group means differ from one another following a statistically significant ANOVA test. SPSS provides 18 PCPs. The purpose of this study was to determine which PCP has the best power and maintains Type I error control both when assumptions of equal sample size and equal variance are met and when they are violated, so as to provide a single resource for use in introductory and intermediate applied statistics courses. Testing the PCPs with simulated data revealed that only the four tests developed to be used under assumption violations adequately controlled Type I error, so we recommend using one of these procedures. Power results were similar for all four of these tests, but were slightly higher for the Games-Howell test than for the others.

Decentralized temporal independent component analysis: Leveraging fMRI data in collaborative settings

The field of neuroimaging has recently witnessed a strong shift towards data sharing; however, current collaborative research projects may be unable to leverage institutional architectures that collect and store data in local, centralized data centers. Additionally, though research groups are willing to grant access for collaborations, they often wish to maintain control of their data locally. These concerns may stem from research culture as well as privacy and accountability concerns. In order to leverage the potential of these aggregated larger data sets, we require tools that perform joint analyses without transmitting the data. Ideally, these tools would have similar performance and ease of use as their current centralized counterparts. In this paper, we propose and evaluate a new Algorithm, decentralized joint independent component analysis (djICA), which meets these technical requirements. djICA shares only intermediate statistics about the data, plausibly retaining privacy of the raw information to local sites, thus making it amenable to further privacy protections, for example via differential privacy. We validate our method on real functional magnetic resonance imaging (fMRI) data and show that it enables collaborative large-scale temporal ICA of fMRI, a rich vein of analysis as of yet largely unexplored, and which can benefit from the larger-N studies enabled by a decentralized approach. We show that djICA is robust to different distributions of data over sites, and that the temporal components estimated with djICA show activations similar to the temporal functional modes analyzed in previous work, thus solidifying djICA as a new, decentralized method oriented toward the frontiers of temporal independent component analysis.

‘Bosons’ and ‘fermions’ in social and economic systems

We analyze social and economic systems with a hierarchical structure and show that for such systems, it is possible to construct thermostatistics, based on the intermediate Gentile statistics. We show that in social and economic hierarchical systems there are elements that obey the Fermi–Dirac statistics and can be called fermions, as well as elements that are approximately subject to Bose–Einstein statistics and can be called bosons. We derive the first and second laws of thermodynamics for the considered economic system and show that such concepts as temperature, pressure and financial potential (which is an analog of the chemical potential in thermodynamics) that characterize the state of the economic system as a whole, can be introduced for economic systems

Stochastic comparisons and multivariate dependence for the epoch times of trend renewal processes

In this paper we study stochastic comparisons and dependence properties for the epoch and inter-epoch times of trend renewal processes with different trend functions. These results extend some of the results on the non-homogeneous Poisson process in Belzunce et al. (2001, 2003) to trend renewal processes. Some applications of the results to a shock model, a repair process, and a specific class of intermediate order statistics are provided.

On the joint limiting distribution of central and intermediate order statistics

On the joint limiting distribution of central and intermediate order statistics

Providing accurate models across private partitioned data: Secure maximum likelihood estimation

This paper focuses on the privacy paradigm of providing access to researchers to remotely carry out analyses on sensitive data stored behind separate firewalls. We address the situation where the analysis demands data from multiple physically separate databases which cannot be combined. Motivating this work is a real model based on research data on kinship foster placement that came from multiple sources and could only be combined through a lengthy process with a trusted research network. We develop and demonstrate a method for accurate calculation of the multivariate normal likelihood, for a set of parameters given the partitioned data, which can then be maximized to obtain estimates. These estimates are achieved without sharing any data or any true intermediate statistics of the data across firewalls. We show that under a certain set of assumptions our method for estimation across these partitions achieves identical results as estimation with the full data. Privacy is maintained by adding noise at each partition. This ensures each party receives noisy statistics, such that the noise cannot be removed until the last step to obtain a single value, the true total log likelihood. Potential applications include all methods utilizing parameter estimation through maximizing the multivariate normal likelihood. We give detailed algorithms, along with available software, and present simulations and analyze the kinship foster placement data estimating structural equation models (SEMs) with partitioned data.

Anyon black holes

We propose a correspondence between an Anyon Van der Waals fluid and a (2+1) dimensional AdS black hole. Anyons are particles with intermediate statistics that interpolates between a Fermi–Dirac statistics and a Bose–Einstein one. A parameter α (0<α<1) characterizes this intermediate statistics of Anyons. The equation of state for the Anyon Van der Waals fluid shows that it has a quasi Fermi–Dirac statistics for α>αc, but a quasi Bose–Einstein statistics for α<αc. By defining a general form of the metric for the (2+1) dimensional AdS black hole and considering the temperature of the black hole to be equal with that of the Anyon Van der Waals fluid, we construct the exact form of the metric for a (2+1) dimensional AdS black hole. The thermodynamic properties of this black hole is consistent with those of the Anyon Van der Waals fluid. For α<αc, the solution exhibits a quasi Bose–Einstein statistics. For α>αc and a range of values of the cosmological constant, there is, however, no event horizon so there is no black hole solution. Thus, for these values of cosmological constants, the AdS Anyon Van der Waals black holes have only quasi Bose–Einstein statistics.

Almost sure limit theorem for the order statistics of stationary Gaussian sequences

In this paper, by using a new comparison inequality for order statistics of Gaussian variables, we proved an almost sure central limit theorem for extreme order statistics of stationary Gaussian sequences with covariance rn under the condition rn log n(log log n)1+? = O(1) for some ? &gt; 0. A similar result on intermediate order statistics is also proved for stationary Gaussian sequences. The obtained results improve some of the existing results.

A strong ergodic theorem for extreme and intermediate order statistics

We study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences. First, we provide sufficient dependence conditions under which these order statistics converges almost surely to the left or right endpoint of the population support, as in the classical setup of sequences of independent and identically distributed random variables. Next, we derive a generalization of this result valid in the class of all strictly stationary sequences. For this purpose, we introduce notions of conditional left and right endpoints of the support of a random variable given a sigma-field, and present basic properties of these concepts. Using these new notions, we prove that extreme and intermediate order statistics from any discrete-time, strictly stationary process converges almost surely to some random variable. We describe the distribution of the limiting variate. Thus we establish a strong ergodic theorem for extreme and intermediate order statistics.

Limit distributions of generalized order statistics in a stationary Gaussian sequence

In this paper we study the limit distributions of extreme, intermediate and central m-generalized order statistics, as well as m-dual generalized order statistics, of a stationary Gaussian sequence under equi-correlated set up. Moreover, the result of extremes is extended to a wide subclass of generalized order statistics, as well as dual generalized order statistics, when the parameters γ1,n, γ2,n, … , γn,n are assumed to be pairwise different.

Generalized-order statistics with random indices

ABSTRACTIn this paper, we study the asymptotic behavior of general sequence of extreme, intermediate and central generalized-order statistics (gos), as well as dual generalized-order statistics (dgos), which are connected asymptotically with some regularly varying functions. Moreover, the limit distribution functions of gos, as well as dgos, with random indices, are obtained under general conditions.

From creation and annihilation operators to statistics

A procedure to derive the partition function of non-interacting particles with exotic or intermediate statistics is presented. The partition function is directly related to the associated creation and annihilation operators that obey some specific commutation or anti-commutation relations. The cases of Gentile statistics, quons, Polychronakos statistics, and ewkons are considered. Ewkons statistics was recently derived from the assumption of free diffusion in energy space (Hoyuelos and Sisterna, 2016); an ideal gas of ewkons has negative pressure, a feature that makes them suitable for the description of dark energy.

A limiting multidimensional distribution of intermediate order statistics

A joint asymptotic distribution of intermediate order statistics constructed from a sample of size n when n→∞ is studied.

Multivariate Order Statistics: the Intermediate Case

Asymptotic normality of intermediate order statistics taken from univariate iid random variables is well-known. We generalize this result to random vectors in arbitrary dimension, where the order statistics are taken componentwise.